If I were a Spanish Major as an undergraduate and decide to pursue a PhD in a completely unrelated field (like Theoretical Physics), it makes sense to give a qualifying exam to check that I had the necessary skills to begin the program. But if I'm coming from a B.S in math to a PhD program also in math, it doesn't seem to make sense to give a qualifying exam, as if the knowledge I gained in my undergraduate was insufficient. I presume that one is accepted into a PhD program because he/she has already demonstrated the "qualifying" skills. Thus, I'm baffled by the notion of the qualifying / prelim exam. I'm curious about the ultimate goals of these exams, and how they relate to the professional development of a graduate student.
Answer
This answer serves mainly to corroborate @Anonymous Mathematician's answer.
As she says, the most important thing to realize is that there are two different kinds of exams that go under the name "prelims / quals". The first of these generally:
(i) tests undergraduate material
(ii) is administered soon after arrival in the graduate program
(iii) used to be used for preliminary weed-out purposes but is now -- at least, in most programs I know about -- used almost entirely for diagnostic purposes.
Probably (iii) is most important: once upon a time, many graduate programs -- even excellent ones, like Berkeley (in fact, especially Berkeley) -- admitted lots of students, as in up to 50% more than were expected to finish. The idea was to give a large group of people, including those with less than sterling (or ivy) pedigrees, a fair shot. Then after a small amount of time in the program -- maybe a year or less -- they would take a "prelim" exam, and a significant portion would fail and leave.
This is no longer the way graduate programs work (at least not in North America, which is what I am primarily familiar with, but to the best of my meager knowledge they don't work that way in other parts of the world either). We pay much closer attention to each student we admit now than in the scenario above, and further our program is judged on retention and completion rates. A graduate program in 2012 who dismissed a third or more of its incoming class every year would look disastrously bad by these sorts of metrics. So this "weedout prelim" is, as far as I know, a thing of the past.
In the graduate program at UGA we still give a "prelim exam" to all entering graduate students, but as I said above we use it almost entirely for diagnostic purposes. In fact we have a certain graduate course designed entirely for students who didn't do well on the prelim, whose purpose is to shore up their undergraduate knowledge ASAP. Other than being encouraged to take this course, there are no direct consequences of failing the prelim (in fact, I'm not sure that one "passes" or "fails" the prelim in any technical sense).
In contrast, most of the "qualifying exams" that you hear graduate students talking about are something entirely different. They:
(i) test graduate level material; in particular, most students do not enter equipped with the knowledge to pass most qualifying exams.
(ii) occupy students' attention for a while: in our program, students have up to three years to pass their qualifying exams.
(iii) really must be passed in order for students to advance in the program, in most cases.
I hope this answers your question. Let me say though that the scores on the "prelim" exam -- i.e., the undergraduate level exam that I mentioned first -- are often all over the place. All of our entering students have at least an undergraduate degree in mathematics. So, unfortunately, no, an undergraduate degree in mathematics is not a guarantee of ability to do undergraduate level mathematics...at least not to the satisfaction of a decent mathematics graduate program. (And a student who does poorly on this entering prelim may yet succeed in doing PhD level mathematics a little later on: that is, the fault often seems to lie with the undergraduate program more than the student.)
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